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Measures – cumulative incidence• Different methods for calculating
– Simple cumulative– Actuarial– Kaplan-Meier– Density
Measures – cumulative incidence• Subscript notation
– R(t0,tj) – risk of disease over the time interval t0 (baseline) to tj (time j)
– R(tj-1,tj) – risk of disease over the time interval tj-1 (time before time j) to tj (time j)
Measures – cumulative incidence• Subscript notation
– N’0 – number at risk of disease at t0 (baseline)– N’0j – number at risk of disease at the beginning of
interval j
Measures – cumulative incidence• Subscript notation
– Ij – incident cases during the interval j– Wj – withdrawals during the interval j
Measures – cumulative incidenceSimple cumulative method:
R(t0,tj) = CI(t0,tj) =
I N'0• Risk calculated across entire study period assuming all
study participants followed for the entire study period, or until disease onset– Assumes no death from competing causes, no withdrawals
• Only appropriate for short time frame
Measures – cumulative incidenceSimple cumulative method:• Example: incidence of a foodborne illness if all those
potentially exposed are identified
Measures – cumulative incidenceActuarial method:
R(tj-1, tj) = CI(tj-1, tj) = IjN'0j - Wj/2
• Risk calculated accounting for fact that some observations will be censored or will withdraw
• Assume withdrawals occur halfway through each observation period on average
• Can be calculated over an entire study period– R(t0,tj) = CI(t0, tj) = I/(N’0-W/2)
• Typically calculated over shorter time frames and risks accumulated
Measures – cumulative incidence
Modification of Szklo Fig. 2-2 – participants observed every 2 months (vs 1)
• Where to start – set up table with time intervals• Fill incident disease cases and withdrawals into appropriate
intervals• Fill in population at risk
Measures – cumulative incidenceActuarial Method
• Calculate interval risk• R(tj-1, tj) = Ij/(N’0j-(Wj/2))
• R(0,2)=1/(10-(1/2)) = 0.11
Measures – cumulative incidenceActuarial Method
• Calculate interval survival• next step: S(tj-1,tj) = 1-
R(tj-1,tj)
Measures – cumulative incidenceActuarial Method
• Calculate cumulative risk – example of time 0 to 10• R(t0, tj) = 1 - Π (1 – R(tj-1,tj)) = 1 - Π (S(tj-1,tj))• R(0, 10) = 1 – (0.89 x 0.88 x 1.0 x 1.0 x 0.85) = 0.34
Measures – cumulative incidenceActuarial Method
• Calculate cumulative survival• S(t0,tj) = 1-R(t0,tj)
Measures – cumulative incidenceActuarial Method
• Intuition for why R(t0, tj) = 1 - Π (Sj) using conditional probabilities
• Example of 5 time intervals:– Π (Sj) = P(S1)*P(S2|S1)*P(S3|S2)*P(S4|S3)*P(S5|S4)
= P(S5)– Multiply first two terms: P(S2|S1)*P(S1) =
P(S2)– Multiplying conditional probabilities gives you
unconditional probability of surviving up to any given time point
– the value (1 - survival) up to (or at) a given time point is then the probability of not surviving up to that time point
Measures – cumulative incidence
Measures – cumulative incidence• Exercise for home (discuss in lab)
– Study population observed monthly for 6 months– Calculate the cumulative incidence of disease from
month 0 to 6
Measures – cumulative incidenceKaplan-Meier method:
IjNj
Rj = CIj =
• Risk calculated at the time each disease event occurs– Accounts for withdrawals in that Nj only includes those at risk at
each time j point– Result differs from actuarial approach in that the time of a
withdrawal (in Kaplan-Meier analysis) coincides with time of an incident disease
• Risks at each onset time j accumulated
• Where to start – set up table with times of incident cases• Fill in population at risk – anyone who has withdrawn by a time j is
no longer at risk at that time
Measures – cumulative incidenceKaplan-Meier Method
JC: discuss withdrawals
• Calculate risk at time j• Rj = Ij/Nj
• R2=1/10 = 0.10• R4=1/8 = 0.125
Measures – cumulative incidenceKaplan-Meier Method
• Survival calculated as in actuarial method• Cumulative risk calculated as in actuarial method
– R(t0, tj) = 1 - Π (1 – Rj) = 1 - Π (Sj)• Cumulative survival calculated as in actuarial method
Measures – cumulative incidenceKaplan-Meier Method
JC: mention product-limit
Measures – cumulative incidenceDensity method:
R (-ID*Δt)
(t0,t) = 1 – S(t0,t) = 1- e
• Depends on functional relationship between a risk and a rate
• Can be calculated over an entire study period if the rate is constant
• Can also be calculated over shorter time frames and risks accumulated
JC: Mention Elandt-Johnson article
Where to start – set up table with time intervals• Fill incident disease cases, withdrawals and population at risk by
interval• Calculate person time (for example used formula PTj=(N’0j-(Wj/2))
Δtj)• Calculate IDj = Ij/PTj
Measures – cumulative incidenceDensity Method
R(t0,t) = 1 – S(t0,t) = 1- e (-ID*Δt)• Calculate interval risk•• R(0,2) = 1-e (-0.05*2) = 0.10
Measures – cumulative incidenceDensity Method
R(t0,t) = 1- e
• Calculate cumulative risk – example of time 0 to 10• Accumulate interval risks as in actuarial method• Or calculate cumulative risk directly
• (-∑ID*Δt)
•R(0,10) = 1-e (-(0.05*2+0.06*2+0*2+0*2+0.08*2) = 0.32
Measures – cumulative incidenceDensity Method
• Cumulative survival calculated as in actuarial method
Measures – cumulative incidenceDensity Method
Measures – cumulative incidenceCumulative incidence• Summary of methods for calculating and basis of
choosing– Simple cumulative – complete follow-up– Actuarial – incomplete follow-up– Kaplan-Meier – incomplete follow-up– Density – converting incidence density to cumulative
incidence
Choosing among the CI methods
• Do you only have rate data? Generally you will choose incidence density.•• Do you have zero withdrawals and a short time period of interest? If so,
simple CI usually OK.•• Do you have fairly exact data on time of incidence and time of withdrawal?
If so, density preferable.•• Do you have fairly exact data on time of incidence but only interval data on
withdrawals? If so, KM most common choice; actuarial or density may not be too different depending on withdrawal timing.
•• Do you have interval data for incidence and withdrawal? If so, actuarial
most common choice, KM and density may not be too different depending on withdrawal timing.
• Assumptions– Uniformity of events and losses within each interval
(the W/2)– Independence between censoring and survival –
otherwise biased/not accurate (also true for ID)– Lack of secular trends
Measures – cumulative incidence
Epidemiologic measures
Szklo Exhibit 2-1